Evenly distributing points on a sphere

Thanks for the kind words. I’m glad you found it interesting and useful.

I think one of the advantages of writing blog posts rather than academic articles is that they are often more readable to a wider audience as the authors can be a little less formal in tone, expand on things (including copious illustrations), without worrying about space constraints.

Hi I included about a dozen references at the end of my post for overall material on this topic. I specifically chose these ones as they are very readable. My favourite paper is defintiely the first one, "A comparison of popular point configurations on S2". It is very comprehensive in breadth as well as historical developments and includes copious references.

One of my other references is "Distributing many points on a sphere" as it is written by E.B. Saff who is basically a legend in this field. Hope that helps! Martin

The values for the tetrahedron, cube, octahedron, dodecahedron and icosahedron are:3.27, 3.27, 3.46,3.19 and 3.64, resp.

These produce the largest d for N=4,8,6,12 and 20 resp. Thus, it is presumed by almost everyone that d=3.64 is the global upper bound.

Unfortunately, I believe it is still an open problem to to prove this or to describe a general upper bound for specific N not equal to any of these five values.

One of the reasons why this field is still so active is because different measures and methods often result in slightly different optimal configurations. In the case of this post, I focused on direct construction methods; and measures relating to minimum distance or convex hulls.

For an excellent commentary on the latest for optimal Riesz energy (which includes Coulomb potentials) configurations can be found in the first paper that I reference: "A comparison of popular point configurations on S2", by Hardin, Michaels and Saff.

Interesting question... I just calculated that for the modified Fibonacci sequence (last method): 2.95 < d_N < 3.12.

Technically d* does not exist, because as N-> infinity, d* alternates between 3.03 and 3.07 (depending on if k is odd or even).

Compare this to the canonical fibonacci sequence gives a value of d_N = 3.07 for all values of N, and so d* = 3.07

Lol and Holy shit! That totally explains why the s2 cell lengths vary based on where they are located in one of the six faces (aka, sides of a cube) of which the documentation does not explain clearly. I have spent many hours using this library and your comment has explained so much of what I was confused about it.

Thank you